32 problems

Let the space of all systems be endowed with its natural topology, and call a system a finite-attractor system when it has finitely many attractors. Palis's density conjecture. Sys…

Let a generic family of maps depend on finitely many parameters, and consider the subset of parameter space corresponding to systems with infinitely many sinks or sources. Finite-a…

Palis's conjecture. The property that a vector field has a large set of trajectories going to its attractors holds for a dense set of vector fields on any compact manifold.

Let $C^r([0,1])$ denote the space of $C^r$ interval maps. For a generic $k$-parameter family $(f_t)_{t\in[-1,1]^k} \subset C^r([0,1])$, where $t\in[-1,1]^k$ is equipped with Lebesg…

Pugh–Shub conjecture. For every $d\geq 1$, a $d$-Kolmogorov typical diffeomorphism displays finitely many sinks and other attractors.

Let the attractor be generated by a gradient system with a global Lyapunov function, and suppose that all equilibria are hyperbolic. Denote its Hausdorff dimension by…

Let $({\mathbb X},d)$ be a metric space, and call $[x,y]:=\{z\in{\mathbb X}:d(x,z)+d(z,y)=d(x,y)\}$ the segment with ends $x,y$. A set is metrically convex if it contains $[x,y]$ f…

Let $F_t$ be a one-parameter iterated function system satisfying properties (H1), (H2), and (H3) of Section 4. If $A_{\bullet}=A^{\bullet}$ for some upper transition attractor of…

Let a sectional-hyperbolic attracting set be $C^1$ close to a sectional-hyperbolic attractor, with no dimensional restrictions; equivalently, allow any combination of stable and ce…

Consider the degenerate wave equation … Here $u$ is the solution, $k\lVert u_t\rVert^p u_t$ is the degenerate damping term, $f$ is the nonlinearity, $K$ is the kernel of the nonloc…

Let $A$ be an attractor of a flow $\varphi$, let $\mu(A)$ denote its intensity under the flow, and let $\mu_t(A)$ denote the intensity of the attractor under the time-$t$ map…

Consider Bianchi type VIII and IX models in the subcritical and critical cases $v\in(0,1/2]$, and let ${\cal A}_-$ denote the attractor as $\tau_-\to\infty$. Define the invariant s…

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with stable set $S\subset\mathrm{K}^{\ocircle}$, attractor ${\cal A}_-$, and $\tau_-\to\infty$. Su…

Let two planar flows each have an attracting homoclinic saddle loop, and consider their direct product. The loops are the attracting homoclinic saddle loops of the respective facto…

Let $G$ be a Lie symmetry group and consider a $G$-invariant Hamiltonian nonlinear PDE … where $\Psi(\cdot,t)$ belongs to $L^2(\mathbb R^3)\otimes\mathbb R^d$,…

Fractal exponential attractor conjecture. The evolution $(T_t,\mathbf H)$ has a fractal exponential attractor, and

Let $\mathcal{V}$ be the parameter space of finite-type double rotations with a fixed partition, and let the associated map have an attractor. The attractor is semi-continuous with…

Let $\mathcal{W}$ be the space of translation vectors for which the associated piecewise translation map with fixed partition $\mathcal{B}$ is of finite type, and let $X_a$ be the…

Let $M$ be a compact manifold. A property is Arnold–Kolmogorov typical if it holds for Lebesgue almost every small parameter in a Baire generic set of smooth families of dynamics.…

Let $f=f_B$, and let $k$ be sufficiently large depending on the internal parameters $Omega$, $L$, $b$, and $p$. There is a proper fractal exponential attractor … with finite fract…

The nonlinear plate dynamics has a fractal exponential attractor constructed in the preceding analysis. Smoothness conjecture. The fractal exponential attractor is in fact smooth.…

Let $B_0$ be the excluded set of initial conditions and let the chaotic invariant set be the hyperbolic invariant set produced by the Smale–Birkhoff homoclinic theorem. Attraction…

Let $H$ be the phase space, let $H^2$ denote the more regular space, let $F$ be the non-linearity satisfying the reasonable assumptions considered in the paper, and let…

The full flow-plate system generates trajectories in the phase space with attractor $\mathbf{A}$. Convergence-to-equilibria conjecture. Individual trajectories of the full flow-pla…

Let $T$ be the dynamical map, let $p$ be the hyperbolic fixed point described in the paper, and let $W^u(p)$ denote its unstable manifold. Let $\Lambda$ be the attractor containing…